Geometrik ölçü teorisi ve ikiz kabarcık zanı

In this thesis we will study the geometric measure theory and the double bubble conjecture. Our treatment of the geometric measure theory will be introductory and yet the theorems and proofs will be studied thoroughly. We will then, as an application of the geometric measure theory, present the double bubble conjecture in R2 and the double bubble conjecture in R3.The geometric measure theory is a branch of differential geometry which deals with maps and surfaces that are not necessarily smooth. It extends the notions of differential geometry with the use of measure theory. One of the most important problems of the geometric measure theory, that has served as the starting point of this field, is to find a spanning set of a given boundary in the Euclidean space with the least area and to decide whether this set has any geometric significance and whether it is unique.The double bubble conjecture in R2 asserts that the standard double bubble in R2 is the unique perimeter minimizing enclosure of two given quantities of area in R2. The double bubble conjecture in R2 has been proven jointly by J. Foisy, M. Alfaro, J. Brock, N. Hodges, and J. Zimba. We will give a complete proof of this conjecture and before giving a sketch of the proof of the double bubble conjecture in R3 we will study the structure theorem which has been proven by M. Hutchings. The structure theorem asserts that except for the standard double bubble there is only one hypersurface that may be a candidate for an area minimizing set enclosing two quantities of volume in R3. M. Hutchings, F. Morgan, M. Ritore, and A. Ros, by using an original stability argument, have ruled out the possibility of any minimizer other than the standard double bubble and hence they have showed that the standard double bubble is the unique area minimizing double bubble enclosing and separating two given quantities of volume in R3.